The **ideal class group** of a number field $K$ with ring of integers $O_K$ is the group of equivalence classes of ideals, given by the quotient of the multiplicative group of all fractional ideals of $O_K$ by the subgroup of principal fractional ideals.

Since $K$ is a number field, the ideal class group of $K$ is a finite abelian group, and so has the structure of a product of cyclic groups encoded by a finite list $[a_1,\dots,a_n]$, where the $a_i$ are positive integers with $a_i\mid a_{i+1}$ for $1\le i<n$.

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2021-04-03 20:28:41

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**History:**(expand/hide all)

- 2021-04-03 20:28:41 by Andrew Sutherland (Reviewed)
- 2019-04-26 14:26:25 by Holly Swisher (Reviewed)
- 2018-08-07 18:55:06 by John Jones

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